Reflexion and Refraction. 133 



by the same cause, whatever it is, which produces a change of 

 phase in metallic reflexion. 



It will be proper to conclude this Essay with a brief sketch 

 of the researches of Sir David Brewster and M. Seebeck, the 



the reflected ray, and As for the change of phase in the refracted ray. Let the 

 same symbols, marked with accents, be used in the second case with similar signi- 

 fications. Then if the incident transversal be taken for unity, we shall have the 

 following formulae : 



1. "When the incident transversal is in the plane of incidence, 



, = M 2 + /t 2 - 2Jf /> cos X 1 



~ M 2 + /* 2 + 2M fji cos x 



^ = M* + ^ + 2#> coT? 

 2Jf u. sin Y 



tanA 2 = 





Mu ^-M Trr^ o > *** Ttir ' 



M * p. A M. + p. COS X ^ 



2. When the incident transversal is perpendicular to the plane of incidence, 



1 + M 2 p? + 2M p cos x' 



~ 1 + Jf V a + 2Jf A* cos 



2Jf u sin Y sin Y 



tan A' 3 = W^ *> tan A'z = -^ - = - . 



M 2 /i2 - 1 Jf^t + cos x J 



"When x = 0, there is no change of phase, and the formulae become identical 

 with those given in the note, p. 101. When x = 90, there is total reflexion at 

 all incidences. The case of pure silver approximates to this. For good speculum 

 metal, x is about 70. The value of M ranges from 2 to 5 in different metals. 



When the incident transversal is inclined to the plane of incidence, its compo- 

 nents, parallel and perpendicular to that plane, will give two reflected transversals 

 with a difference of phase equal to A'a - As. The reflected vibration will then be 

 performed in an ellipse ; and the position and magnitude of the axes of the ellipse 

 may be deduced from the preceding formulae. The consequences of these formulae 

 are very simple and elegant, but I cannot dwell upon them here. Suffice it to 

 observe, that every angle of incidence has another angle corresponding to it, which 

 I call its conjugate angle of incidence ; and that the value of A'a - AS at one of 

 these angles is the supplement of its value at the other, while the ratio 3 is the 

 same at both angles ; whence it follows that, ceteris paribus, the elliptic vibrations, 

 reflected at conjugate angles, are similar to each other, and have their homologous 



